Domain Theory and the Causal Structure of Spacetimeprakash/Talks/cie08.pdf · Domain Theory Domains and Causal Structure Interval Domains Reconstructing spacetime Domain Theory and

IntroductionCausal Structure

Domain TheoryDomains and Causal Structure

Interval DomainsReconstructing spacetime

Domain Theory and the Causal Structure ofSpacetime

Keye Martin1 and Prakash Panangaden2

1Center for High Assurance SystemsNaval Research Laboratory

2School of Computer ScienceMcGill University

19th June 2008 / Computability in Europe.

Panangaden Domain Theory and the Causal Structure of Spacetime




Outline

1 Introduction

2 Causal Structure

3 Domain Theory

4 Domains and Causal Structure

5 Interval Domains

6 Reconstructing spacetime





Outline

1 Introduction

2 Causal Structure

3 Domain Theory


5 Interval Domains






Outline

1 Introduction

2 Causal Structure

3 Domain Theory


5 Interval Domains






Outline

1 Introduction

2 Causal Structure

3 Domain Theory


5 Interval Domains






Outline

1 Introduction

2 Causal Structure

3 Domain Theory


5 Interval Domains






Outline

1 Introduction

2 Causal Structure

3 Domain Theory


5 Interval Domains






Overview

Causality is taken to be fundamental in physics and equallyin computer science.

In spacetime physics: Newton, Einstein, Penrose,Hawking, Sorkin.

In computer science: Petri, Lamport, Pratt, Winskel.

Causality forms a partial order: there are no causal cycles.

Ordered topological spaces (domains) were used by DanaScott and Yuri Ershov to model computation as informationprocessing.

Spacetime carries a natural domain structure.





Overview











Overview











Overview











Overview











Overview











Motivation

We do not know how to reconcile relativity and quantummechanics.

The holy grail: a theory of quantum gravity.

One approach [Sorkin] : discrete causal sets. Spacetime isa discrete poset. Causality is the fundamental structure.

Our work is inspired by Sorkin but it is not the sameframework.

We are exploring the classical analogue of this question: innon-quantum relativity can the causal order be taken asfundamental?





Motivation










Motivation










Motivation










Motivation










Where Does Computability Come In?

Scott’s vision: computability should be continuity in sometopology.

A finite piece of information about the output should onlyrequire a finite piece of information about the input.

This is just what the ǫ − δ definition says.

Data types are domains (ordered topological spaces) andcomputable functions are continuous.
































Summary of Results

The causal order alone determines the topology of globallyhyperbolic spacetimes. [CMP Nov’06]

A (globally hyperbolic) spacetime can be given domainstructure: approximate points. [CMP Nov’06]

The space of causal curves in the Vietoris topology iscompact. (cf. Sorkin-Woolgar) [GRG ’06]

The geometry can be captured by a Martin “measurement.”[in prep.]





Summary of Results









Summary of Results









Summary of Results









The first two items actually work for strongly causal spacetimes(Sumati Surya, in prep.).





The Newtonian View

Spacetime is a 4-dimensional manifold M.

M has a canonical product structure: M = R × S. R is“absolute” time and S is space.

An event at (t , ~x) can influence any other event (t ′, ~x′) ift < t ′.

“Now” (a surface of simultaneity) is the boundary betweenthe past and the future and defines “space.”





The Newtonian View









The Newtonian View









The Newtonian View









A Picture of Newtonian Spacetime

Now

Future

Past





The layers of spacetime structure

Set of events: no structure

Topology: 4 dimensional real manifold, Hausdorff,paracompact,...

Differentiable structure: tangent spaces

Causal structure: light cones, defines metric up toconformal transformations. This is 9

10 of the metric.

Parallel transport: affine structure.

Lorentzian metric: gives a length scale.










10 of the metric.












10 of the metric.












10 of the metric.












10 of the metric.












10 of the metric.







The causal structure of spacetime

At every point a pair of “cones” is defined in the tangentspace: future and past light cone. A vector on the cone iscalled null or lightlike and one inside the cone is calledtimelike.

We assume that spacetime is time-orientable: there is aglobal notion of future and past.

A timelike curve from x to y has a tangent vector that iseverywhere timelike: we write x ≺ y . (We avoid x ≪ y fornow.) A causal curve has a tangent that, at every point, iseither timelike or null: we write x ≤ y .





















Relativistic Spacetime





Causal Structure of Spacetime II

I+(x) := {y ∈ M|x ≺ y}; similarly I−

J+(x) := {y ∈ M|x ≤ y}; similarly J−.

I± are always open sets in the manifold topology; J± arenot always closed sets.

Chronology: x ≺ y ⇒ y 6≺ x .

Causality: x ≤ y and y ≤ x implies x = y .













































Fundamental Causality Assumption

≤ is a partial order.





J+(x) and I−(x)

excludesboundary

J+(x)

I−(x)

x





Causality Conditions

I±(p) = I±(q) ⇒ p = q.

Strong causality at p: Every neighbourhood O of pcontains a neighbourhood U ⊂ O such that no causalcurve can enter U , leave it and then re-enter it.

Stable causality: perturbations of the metric do not causeviolations of causality.

Causal simplicity: for all x ∈ M, J±(x) are closed.






I±(p) = I±(q) ⇒ p = q.









I±(p) = I±(q) ⇒ p = q.









I±(p) = I±(q) ⇒ p = q.








Global Hyperbolicity

Spacetime has good initial data surfaces for globalsolutions to hyperbolic partial differential equations (waveequations). [Leray]

Global hyperbolicity: M is strongly causal and for each p, qin M, [p, q] := J+(p) ∩ J−(q) is compact.





Global Hyperbolicity

Spacetime has good initial data surfaces for globalsolutions to hyperbolic partial differential equations (waveequations). [Leray]

Global hyperbolicity: M is strongly causal and for each p, qin M, [p, q] := J+(p) ∩ J−(q) is compact.





A Spacetime Interval

x

y

J+(x) ∩ J−(y)





The Alexandrov Topology

Define〈x , y〉 := I+(x) ∩ I−(y).

The sets of the form 〈x , y〉 form a base for a topology on Mcalled the Alexandrov topology.Theorem (Penrose): TFAE:

1 (M, g) is strongly causal.2 The Alexandrov topology agrees with the manifold

topology.3 The Alexandrov topology is Hausdorff.






Define〈x , y〉 := I+(x) ∩ I−(y).









Define〈x , y〉 := I+(x) ∩ I−(y).








Recursion as a Fixed Point

Kleene had the idea of explaining recursion as a fixedpoint.

Scott: how to obtain a model of the λ-calculus?

D ≡ [D → D].

No way!!

Construct a model of the λ-calculus using posets andtopologies on posets.

Use the topology to cut down to the continuous maps.








D ≡ [D → D].

No way!!










D ≡ [D → D].

No way!!










D ≡ [D → D].

No way!!







Domain theory

Order as (qualitative) information content.

data types are organized into so-called “domains”:directed-complete (directed sets have least upper bounds)posets

For “directed set” think “chain.”

computable functions are viewed as continuous withrespect to a suitable topology: the Scott topology.

ideal (infinite) elements are limits of their (finite)approximations.





Domain theory










Domain theory










Domain theory










Domain theory










Examples of domains

The integers with no relation between them and a specialelement ⊥ below all the integers: a flat domain.

Sequences of elements from {a, b} ordered by prefix: thedomain of streams.

Compact non-empty intervals of real numbers ordered byreverse inclusion (with R thrown in).

X a locally compact space with K (X ) the collection ofcompact subsets ordered by reverse inclusion.





Examples of domains









Examples of domains









Examples of domains









Computation on topological spaces

One can view domain theory as a way of formalizing atheory of computability on topological spaces: Tucker andStoltenberg-Hansen.

Open sets are finitely checkable properties.

More generally, one can seek a domain theoretic analogueof analysis seeking a computational handle on the subject:Edalat, Escardo, Alex Simpson, Keye Martin...





















Computational Intuition: Streams

A box with input streaming in and output streaming out:may proceed forever.

The more we see the better we know the “complete”output.

Anything that is output cannot be retracted.

Computable: to see a finite “portion” of the output only afinite amount of input can be required.

Typical non-computable function: output “yes” if there areinfinitely many zeros in the input.













































The Way-below relation

In addition to ≤ there is an additional, (often) irreflexive,transitive relation written ≪: x ≪ y means that x has a“finite” piece of information about y or x is a “finiteapproximation” to y . If x ≪ x we say that x is finite.

The relation x ≪ y - pronounced x is “way below” y - isdirectly defined from ≤.

Official definition of x ≪ y : If X ⊂ D is directed andy ≤ (⊔X ) then there exists u ∈ X such that x ≤ u. If a limitgets past y then some finite stage of the limiting processalready got past x .





















Domain theory continued

A continuous domain D has a basis of elements B ⊂ Dsuch that for every x in D the set x ⇓:= {u ∈ B|u ≪ x} isdirected and ⊔(x ⇓) = x .

A continuous function between domains is order monotoneand preserves lubs (sups) of directed sets.

Why are directed sets so important? They are collectingconsistent pieces of information.

Surely the words “continuous function” should havesomething to do with topology?
































The dream

Find a topology so that computability is precisely continuity.

Scott’s topology captures some aspects of computabilitybut not all.

All computable functions are Scott continuous but notconversly.

One still needs recursion theory to pin down exactly whatcomputable means: Gordon Plotkin, Mike Smythe andeffectively given domains





The dream









The dream









The dream









Topologies of Domains 1: The Scott topology

O ⊆ D is Scott open if it is upwards closed and

if X ⊂ D and ⊔X ∈ O it must be the case that some x ∈ Xis in O.

The effectively checkable properties.

This topology is T0 but not T1.
































Topologies of Domains 2: The Lawson topology

basis of the formO \ [∪i(xi ↑)].

Says something about negative information.

This topology is metrizable.

It has the same Borel algebra as the Scott topology.
































Topologies of Domains 3: The interval topology

Basis sets of the form [x , y ] := {u|x ≪ u ≪ y}.

The domain theoretic analogue of the Alexandrov topology.

Caveat: the “Alexandrov topology” means something elsein the theory of topological lattices.





















The role of way below in spacetime structure

Theorem: Let (M, g) be a spacetime with Lorentziansignature. Define x ≪ y as the way-below relation of thecausal order. If (M, g) is globally hyperbolic then x ≪ y iffy ∈ I+(x).

One can recover I from J without knowing what smooth ortimelike means.

Intuition: any way of approaching y must involve gettinginto the timelike future of x .





















We can stop being coy about notational clashes: henceforth ≪is way-below and the timelike order.





Bicontinuity and Global Hyperbolicity

The definition of continuous domain - or poset - is biasedtowards approximation from below. If we symmetrize thedefinitions we get bicontinuity (details in the CMP paper).

Theorem: If (M, g) is globally hyperbolic then (M,≤) is abicontinuous poset.

In this case the interval topology is the manifoldtopology.





















An “abstract” version of globally hyperbolic

We define a globally hyperbolic poset (X ,≤) to be1 bicontinuous and,2 all segments [a, b] := {x : a ≤ x ≤ b} are compact in the

interval topology on X .





An “abstract” version of globally hyperbolic

We define a globally hyperbolic poset (X ,≤) to be1 bicontinuous and,2 all segments [a, b] := {x : a ≤ x ≤ b} are compact in the

interval topology on X .





An Important Example of a Domain: IR

The collection of compact intervals of the real line

IR = {[a, b] : a, b ∈ R & a ≤ b}

ordered under reverse inclusion

[a, b] ⊑ [c, d ] ⇔ [c, d ] ⊆ [a, b]

is an ω-continuous dcpo.

For directed S ⊆ IR,⊔

S =⋂

S.





An Important Example of a Domain: IR

The collection of compact intervals of the real line

IR = {[a, b] : a, b ∈ R & a ≤ b}

ordered under reverse inclusion

[a, b] ⊑ [c, d ] ⇔ [c, d ] ⊆ [a, b]

is an ω-continuous dcpo.

For directed S ⊆ IR,⊔

S =⋂

S.





IR continued.

I ≪ J ⇔ J ⊆ int(I), and

{[p, q] : p, q ∈ Q & p ≤ q} is a countable basis for IR.





IR continued.

I ≪ J ⇔ J ⊆ int(I), and

{[p, q] : p, q ∈ Q & p ≤ q} is a countable basis for IR.





The domain IR is called the interval domain.

We have max(IR) ≃ R in the Scott topology.

The “classical” structure lives on top - ideal points,

there is now a substrate of “approximate” elements.





























Generalizing IR

The closed segments of a globally hyperbolic poset X

IX := {[a, b] : a ≤ b & a, b ∈ X}

ordered by reverse inclusion form a continuous domainwith

[a, b] ≪ [c, d ] ≡ a ≪ c & d ≪ b.





Generalizing IR

The closed segments of a globally hyperbolic poset X

IX := {[a, b] : a ≤ b & a, b ∈ X}

ordered by reverse inclusion form a continuous domainwith

[a, b] ≪ [c, d ] ≡ a ≪ c & d ≪ b.





max(IX ) ≃ X

where the set of maximal elements has the relative Scotttopology from IX .





Spacetime from a discrete ordered set

If we have a countable dense subset C of M, a globallyhyperbolic spacetime, then we can view the induced causalorder on C as defining a discrete poset. An ideal completionconstruction in domain theory, applied to a poset constructedfrom C yields a domain IC with

max(IC) ≃ M

where the set of maximal elements have the Scott topology.Thus from a countable subset of the manifold we canreconstruct the manifold as a topological space.





Globally Hyperbolic Posets and Interval Domains

One can define a category of globally hyperbolic posets

and an abstract notion of “interval domain”: these can alsobe organized into a category.

These two categories are equivalent.

Thus globally hyperbolic spacetimes are domains - not justposets - but

not with the causal order but, rather, with the order comingfrom the notion of intervals; i.e. from notions ofapproximation.













































Spacetime as a domain

The domain consists of intervals [x , y ] = J+(x) ∩ J−(y).

For globally hyperbolic spacetimes these are all compact.

The order is inclusion.

The maximal elements are the usual pointsx = J+(x) ∩ J−(x).

The other elements are “approximate points.”













































Other layers of structure

We would like to put differential structure on the domainand

metric structure as well.

There are derivative concepts for domains – not yetexplored in this context.

Keye Martin defined a concept called a “measurement.”This is designed to capture quantitative notions ondomains.

Metric notions can be related to these measurements.













































Keye’s measurements

A measurement on D is a function µ : D → (∞, 0] (reverseordered) that is Scott continuous and satisfies some extraconditions.

We write ker(µ) for {x |µ(x) = 0} andµǫ(x) = {y |y ⊑ x and |µ(x) − µ(y)| ≤ ǫ}.

For any Scott open set U and any x ∈ ker(µ)

x ∈ U ⇒ (∃ǫ > 0)x ∈ µǫ ⊆ U.









x ∈ U ⇒ (∃ǫ > 0)x ∈ µǫ ⊆ U.









x ∈ U ⇒ (∃ǫ > 0)x ∈ µǫ ⊆ U.





Idea: µ(x) measures the “uncertainty” in x .

Maximal elements have zero uncertainty.

Measurements explain why some non-continuousfunctions have fixed points.

Example: the bisection algorithm.





























Measurements and Geometry

Does the volume of an interval or the length of the longestgeodesic give a measurement on the domain of spacetimeintervals?

Unfortunately not! If a and b are null related then you get anontrivial interval with zero volume.

However, any globally hyperbolic spacetime (in fact anystably causal one) has a global time function. Thedifference in the global time function does give ameasurement.

Knowing the global time function effectively gives the restof the metric.
































Conclusions

Domain theoretic methods are a fruitful way of exploringthe properties of causal structure.

The fact that globally hyperbolic posets are intervaldomains gives a sensible way of thinking of“approximations” to spacetime points in terms of intervals.

We would like to relate our approximate points to Sorkin’sdiscrete spacetimes.

How do we understand differential structure in thisframework? This is vital for understanding how to describedynamics of systems living on spacetime.





Conclusions









Conclusions









Conclusions









Thanks! and welcome back to Earth.


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Domain Theory and the Causal Structure of Spacetimeprakash/Talks/cie08.pdf · Domain Theory Domains and Causal Structure Interval Domains Reconstructing spacetime Domain Theory and